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== '' ? this.value = 'Search' : true" /></fieldset> </form> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="model_category_theory">Model category theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>, <a class="existingWikiWord" href="/nlab/show/model+%28infinity%2C1%29-category">model <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-category</a> Definitions <ul> <li> <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a> (<a class="existingWikiWord" href="/nlab/show/relative+category">relative category</a>, <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/fibration">fibration</a>, <a class="existingWikiWord" href="/nlab/show/cofibration">cofibration</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/weak+factorization+system">weak factorization system</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/resolution">resolution</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+%28as+an+operation%29">homotopy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">of a model category</a> </li> </ul> Morphisms <ul> <li> <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Quillen+bifunctor">Quillen bifunctor</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a> </li> </ul> </li> </ul> Universal constructions <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a> <a class="existingWikiWord" href="/nlab/show/homotopy+weighted+colimit">homotopy weighted (co)limit</a> <a class="existingWikiWord" href="/nlab/show/homotopy+coend">homotopy (co)end</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bousfield-Kan+map">Bousfield-Kan map</a> </li> </ul> Refinements <ul> <li> <a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/monoidal+Quillen+adjunction">monoidal Quillen adjunction</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/enriched+Quillen+adjunction">enriched Quillen adjunction</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/monoidal+enriched+model+category">monoidal enriched model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/simplicial+Quillen+adjunction">simplicial Quillen adjunction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/simplicial+monoidal+model+category">simplicial monoidal model category</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cellular+model+category">cellular model category</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/algebraic+model+category">algebraic model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/compactly+generated+model+category">compactly generated model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/proper+model+category">proper model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cartesian+closed+model+category">cartesian closed model category</a>, <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+model+category">locally cartesian closed model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a> </li> </ul> Producing new model structures <ul> <li> <a class="existingWikiWord" href="/nlab/show/global+model+structures+on+functor+categories">on functor categories (global)</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+an+overcategory">on slice categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">Bousfield localization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred model structure</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebraic+fibrant+objects">on algebraic fibrant objects</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Grothendieck+construction+for+model+categories">Grothendieck construction for model categories</a> </li> </ul> Presentation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories <ul> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categorical+hom-space">(∞,1)-categorical hom-space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable (∞,1)-category</a> </li> </ul> Model structures <ul> <li><a class="existingWikiWord" href="/nlab/show/Cisinski+model+structure">Cisinski model structure</a></li> </ul> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids <a class="existingWikiWord" href="/nlab/show/model+structure+for+%E2%88%9E-groupoids">for ∞-groupoids</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+topological+spaces">on topological spaces</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+compactly+generated+topological+spaces">on compactly generated spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+Delta-generated+topological+spaces">on Delta-generated spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+diffeological+spaces">on diffeological spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+structure">Strøm model structure</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Thomason+model+structure">Thomason model structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+over+a+test+category">model structure on presheaves over a test category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">on simplicial sets</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+semi-simplicial+sets">on semi-simplicial sets</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/constructive+model+structure+on+simplicial+sets">constructive model structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+for+right+fibrations">for right/left fibrations</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+groupoids">model structure on simplicial groupoids</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+cubical+sets">on cubical sets</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+strict+%E2%88%9E-groupoids">on strict ∞-groupoids</a>, <a class="existingWikiWord" href="/nlab/show/natural+model+structure+on+groupoids">on groupoids</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes">on chain complexes</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+abelian+groups">model structure on cosimplicial abelian groups</a> related by the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+simplicial+sets">model structure on cosimplicial simplicial sets</a> </li> </ul> for equivariant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids <ul> <li> <a class="existingWikiWord" href="/nlab/show/fine+model+structure+on+topological+G-spaces">fine model structure on topological G-spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/coarse+model+structure+on+topological+G-spaces">coarse model structure on topological G-spaces</a> (<a class="existingWikiWord" href="/nlab/show/Borel+model+structure">Borel model structure</a>) </li> </ul> for rational <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dgc-algebras">model structure on dgc-algebras</a></li> </ul> for rational equivariant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+equivariant+chain+complexes">model structure on equivariant chain complexes</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+equivariant+dgc-algebras">model structure on equivariant dgc-algebras</a> </li> </ul> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-groupoids <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+for+n-groupoids">for n-groupoids</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+for+n-groupoids">for n-types</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/canonical+model+structure+on+groupoids">for 1-groupoids</a> </li> </ul> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groups <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+groups">model structure on simplicial groups</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+reduced+simplicial+sets">model structure on reduced simplicial sets</a> </li> </ul> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-algebras general <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-algebras <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+monoids">on monoids</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">on simplicial T-algebras</a>, on <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a>s </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+a+monad">on algebas over a monad</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">on algebras over an operad</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+modules+over+an+algebra+over+an+operad">on modules over an algebra over an operad</a> </li> </ul> specific <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-algebras <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">model structure on differential-graded commutative algebras</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+differential+graded-commutative+superalgebras">model structure on differential graded-commutative superalgebras</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras+over+an+operad">on dg-algebras over an operad</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">on dg-algebras</a> and on <a class="existingWikiWord" href="/nlab/show/simplicial+ring">on simplicial rings</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+rings">on cosimplicial rings</a> related by the <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">for L-∞ algebras</a>: <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">on dg-Lie algebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">on dg-coalgebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+Lie+algebras">on simplicial Lie algebras</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-modules">model structure on dg-modules</a> </li> </ul> for stable/spectrum objects <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+spectra">model structure on spectra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+ring+spectra">model structure on ring spectra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+parameterized+spectra">model structure on parameterized spectra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+of+spectra">model structure on presheaves of spectra</a> </li> </ul> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+categories+with+weak+equivalences">on categories with weak equivalences</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+quasi-categories">Joyal model for quasi-categories</a> (and its <a class="existingWikiWord" href="/nlab/show/model+structure+for+cubical+quasicategories">cubical version</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-categories">on sSet-categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+for+complete+Segal+spaces">for complete Segal spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">for Cartesian fibrations</a> </li> </ul> for stable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-categories">on dg-categories</a></li> </ul> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-operads <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">on operads</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+Segal+operads">for Segal operads</a> <a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">on algebras over an operad</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+modules+over+an+algebra+over+an+operad">on modules over an algebra over an operad</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+dendroidal+sets">on dendroidal sets</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+dendroidal+complete+Segal+spaces">for dendroidal complete Segal spaces</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+dendroidal+Cartesian+fibrations">for dendroidal Cartesian fibrations</a> </li> </ul> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n,r)</annotation></semantics></math>-categories <ul> <li> <a class="existingWikiWord" href="/nlab/show/Theta+space">for (n,r)-categories as ∞-spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+for+weak+complicial+sets">for weak ∞-categories as weak complicial sets</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+cellular+sets">on cellular sets</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/canonical+model+structure">on higher categories in general</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+strict+%E2%88%9E-categories">on strict ∞-categories</a> </li> </ul> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-sheaves / <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stacks <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+homotopical+presheaves">on homotopical presheaves</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">on simplicial presheaves</a> <a class="existingWikiWord" href="/nlab/show/global+model+structure+on+simplicial+presheaves">global model structure</a>/<a class="existingWikiWord" href="/nlab/show/Cech+model+structure+on+simplicial+presheaves">Cech model structure</a>/<a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+presheaves">local model structure</a> <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sheaves">on simplicial sheaves</a> <a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+of+simplicial+groupoids">on presheaves of simplicial groupoids</a> <a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-enriched+presheaves">on sSet-enriched presheaves</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+for+%282%2C1%29-sheaves">model structure for (2,1)-sheaves</a>/for stacks </li> </ul> </div></div> <h4 id="differential_geometry">Differential geometry</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic</a> <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> Introductions <a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+1">from point-set topology to differentiable manifolds</a> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a>: <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+coordinate+systems">coordinate systems</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+spaces">smooth spaces</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+manifolds+and+orbifolds">manifolds</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+homotopy+types">smooth homotopy types</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">supergeometry</a> Differentials <ul> <li> <a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a>, <a class="existingWikiWord" href="/nlab/show/chain+rule">chain rule</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differentiable+function">differentiable function</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesimal space</a>, <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+point">infinitesimally thickened point</a>, <a class="existingWikiWord" href="/nlab/show/amazing+right+adjoint">amazing right adjoint</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/V-manifolds">V-manifolds</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, <a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a>, <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a>, <a class="existingWikiWord" href="/nlab/show/exotic+smooth+structure">exotic smooth structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/analytic+manifold">analytic manifold</a>, <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/formal+smooth+manifold">formal smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>, <a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/manifold+structure+of+mapping+spaces">manifold structure of mapping spaces</a> </li> </ul> Tangency <ul> <li> <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>, <a class="existingWikiWord" href="/nlab/show/frame+bundle">frame bundle</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a>, <a class="existingWikiWord" href="/nlab/show/multivector+field">multivector field</a>, <a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a>; </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+forms+in+synthetic+differential+geometry">differential forms</a>, <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a>, <a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault complex</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback of differential forms</a>, <a class="existingWikiWord" href="/nlab/show/invariant+differential+form">invariant differential form</a>, <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+form">Maurer-Cartan form</a>, <a class="existingWikiWord" href="/nlab/show/horizontal+differential+form">horizontal differential form</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/cogerm+differential+form">cogerm differential form</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+diffeomorphism">local diffeomorphism</a>, <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale morphism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/submersion">submersion</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+morphism">formally smooth morphism</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/immersion">immersion</a>, <a class="existingWikiWord" href="/nlab/show/formally+unramified+morphism">formally unramified morphism</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/de+Rham+space">de Rham space</a>, <a class="existingWikiWord" href="/nlab/show/crystal">crystal</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/infinitesimal+disk+bundle">infinitesimal disk bundle</a> </li> </ul> The magic algebraic facts <ul> <li> <a class="existingWikiWord" href="/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">embedding of smooth manifolds into formal duals of R-algebras</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/smooth+Serre-Swan+theorem">smooth Serre-Swan theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/derivations+of+smooth+functions+are+vector+fields">derivations of smooth functions are vector fields</a> </li> </ul> Theorems <ul> <li> <a class="existingWikiWord" href="/nlab/show/Hadamard+lemma">Hadamard lemma</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Borel%27s+theorem">Borel's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Boman%27s+theorem">Boman's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Whitney+extension+theorem">Whitney extension theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Steenrod-Wockel+approximation+theorem">Steenrod-Wockel approximation theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Whitney+embedding+theorem">Whitney embedding theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hochschild-Kostant-Rosenberg+theorem">Hochschild-Kostant-Rosenberg theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+cohomology+hexagon">differential cohomology hexagon</a> </li> </ul> Axiomatics <ul> <li> <a class="existingWikiWord" href="/nlab/show/Kock-Lawvere+axiom">Kock-Lawvere axiom</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a>, <a class="existingWikiWord" href="/nlab/show/super+smooth+topos">super smooth topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/microlinear+space">microlinear space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/integration+axiom">integration axiom</a> </li> </ul> </li> </ul> <div> <a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a> <ul> <li> (<a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sharp+modality">sharp modality</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="0em" rspace="thinmathspace">ʃ</mo><mo>⊣</mo><mo>♭</mo><mo>⊣</mo><mo>♯</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\esh \dashv \flat \dashv \sharp )</annotation></semantics></math> <ul> <li> <a class="existingWikiWord" href="/nlab/show/discrete+object">discrete object</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+object">codiscrete object</a>, <a class="existingWikiWord" href="/nlab/show/concrete+object">concrete object</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/points-to-pieces+transform">points-to-pieces transform</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures">structures in cohesion</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/dR-shape+modality">dR-shape modality</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/dR-flat+modality">dR-flat modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">ʃ</mo> <mi>dR</mi></msub><mo>⊣</mo><msub><mo>♭</mo> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">\esh_{dR} \dashv \flat_{dR}</annotation></semantics></math> </li> </ul> <a class="existingWikiWord" href="/nlab/show/infinitesimal+cohesion">infinitesimal cohesion</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+modality">classical modality</a></li> </ul> <a class="existingWikiWord" href="/nlab/show/tangent+cohesive+%28%E2%88%9E%2C1%29-topos">tangent cohesion</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+cohomology+diagram">differential cohomology diagram</a></li> </ul> <a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a> <ul> <li> (<a class="existingWikiWord" href="/nlab/show/reduction+modality">reduction modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+shape+modality">infinitesimal shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+flat+modality">infinitesimal flat modality</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℜ</mi><mo>⊣</mo><mi>ℑ</mi><mo>⊣</mo><mi>&</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Re \dashv \Im \dashv \&)</annotation></semantics></math> <ul> <li> <a class="existingWikiWord" href="/nlab/show/reduced+object">reduced object</a>, <a class="existingWikiWord" href="/nlab/show/coreduced+object">coreduced object</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+object">formally smooth object</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+map">formally étale map</a> </li> <li> <a href="cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#StructuresInDifferentialCohesion">structures in differential cohesion</a> </li> </ul> </li> </ul> <a class="existingWikiWord" href="/nlab/show/super+smooth+infinity-groupoid">graded differential cohesion</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/fermionic+modality">fermionic modality</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/bosonic+modality">bosonic modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/rheonomy+modality">rheonomy modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⇉</mo><mo>⊣</mo><mo>⇝</mo><mo>⊣</mo><mi>Rh</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)</annotation></semantics></math> </li> </ul> <a class="existingWikiWord" href="/nlab/show/orbifold+cohomology">singular cohesion</a> <div id="Diagram" class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>id</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>id</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>fermionic</mi></mover></mtd> <mtd><mo>⇉</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi mathvariant="normal">R</mi><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mi mathvariant="normal">h</mi></mtd> <mtd><mover><mrow></mrow><mi>rheonomic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>reduced</mi></mover></mtd> <mtd><mi>ℜ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>&</mi></mtd> <mtd><mover><mrow></mrow><mtext>étale</mtext></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>cohesive</mi></mover></mtd> <mtd><mo lspace="0em" rspace="thinmathspace">ʃ</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♭</mo></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd> <mtd><mo>♭</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♯</mo></mtd> <mtd><mover><mrow></mrow><mi>continuous</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>∅</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{&#233;tale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast } </annotation></semantics></math></div></div> Models <ul> <li> <a class="existingWikiWord" href="/nlab/show/Models+for+Smooth+Infinitesimal+Analysis">Models for Smooth Infinitesimal Analysis</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/smooth+algebra">smooth algebra</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">C^\infty</annotation></semantics></math>-ring) </li> <li> <a class="existingWikiWord" href="/nlab/show/smooth+locus">smooth locus</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fermat+theory">Fermat theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/formal+smooth+%E2%88%9E-groupoid">formal smooth ∞-groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+formal+smooth+%E2%88%9E-groupoid">super formal smooth ∞-groupoid</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Lie+n-algebra">Lie n-algebra</a>, <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a>, <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-group">smooth ∞-group</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/differential+equations">differential equations</a>, <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/D-geometry">D-geometry</a>, <a class="existingWikiWord" href="/nlab/show/D-module">D-module</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+complex">Euler-Lagrange complex</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation">Euler-Lagrange equation</a>, <a class="existingWikiWord" href="/nlab/show/de+Donder-Weyl+formalism">de Donder-Weyl formalism</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a>, <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">connection on an ∞-bundle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne complex</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/higher+parallel+transport">higher parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+cohomology">fiber integration in differential cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a>, <a class="existingWikiWord" href="/nlab/show/higher+holonomy">higher holonomy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a>, <a class="existingWikiWord" href="/nlab/show/higher+gauge+theory">higher gauge theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wilson+line">Wilson line</a>, <a class="existingWikiWord" href="/nlab/show/Wilson+surface">Wilson surface</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a> (<a class="existingWikiWord" href="/nlab/show/super+Cartan+geometry">super</a>, <a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher</a>) <ul> <li> <a class="existingWikiWord" href="/nlab/show/Klein+geometry">Klein geometry</a>, (<a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a>, <a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion of a G-structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Euclidean+geometry">Euclidean geometry</a>, <a class="existingWikiWord" href="/nlab/show/hyperbolic+geometry">hyperbolic geometry</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+geometry">elliptic geometry</a> </li> <li> (<a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+geometry">pseudo</a>-)<a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian geometry</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/isometry">isometry</a>, <a class="existingWikiWord" href="/nlab/show/Killing+vector+field">Killing vector field</a>, <a class="existingWikiWord" href="/nlab/show/Killing+spinor">Killing spinor</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, <a class="existingWikiWord" href="/nlab/show/super-spacetime">super-spacetime</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/complex+geometry">complex geometry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+geometry">conformal geometry</a> </li> </ul> </div></div> <h4 id="homotopy_theory">Homotopy theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>… models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, … see also <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a> Introductions <ul> <li> <a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a> </li> </ul> Definitions <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> Paths and cylinders <ul> <li> <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/path+object">path object</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a> </li> </ul> </li> </ul> Homotopy groups <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> Basic facts <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> Theorems <ul> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#details'>Details</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> There may be different <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>-<a class="existingWikiWord" href="/nlab/show/mathematical+structure">structures</a> on the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a>. Of practical interest would be a model structure whose <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> are the <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> on standard smooth <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> of <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a>, i.e. the <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> between the <a class="existingWikiWord" href="/nlab/show/cohesive+shapes">cohesive shapes</a> of <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a> regarded as the <a class="existingWikiWord" href="/nlab/show/concrete+objects">concrete objects</a> in the <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> of <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoids">smooth ∞-groupoids</a>. By the discussion at <a class="existingWikiWord" href="/nlab/show/shape+via+cohesive+path+%E2%88%9E-groupoid">shape via cohesive path ∞-groupoid</a> these are detected by smooth functions out of <a class="existingWikiWord" href="/nlab/show/simplices">simplices</a> or <a class="existingWikiWord" href="/nlab/show/cubes">cubes</a> with their canonical diffeological structure (as discussed in <a href="#ChristensenWu14">Christensen-Wu 14</a>). A model category with this property has been claimed in <a href="#HaraguchiShimakawa13">Haraguchi-Shimakawa 13</a>,, <a href="#HaraguchiShimakawa20">Haraguchi-Shimakawa 20</a> Another model category structure is discussed in <a href="#Kihara16">Kihara 16</a>, but this uses a non-standard diffeology on simplices (to enforce that all smooth <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complexes">singular simplicial complexes</a> are <a class="existingWikiWord" href="/nlab/show/fibrant+objects">fibrant objects</a>). <h2 id="details">Details</h2> The approach of <a href="#HaraguchiShimakawa13">Haraguchi-Shimakawa 13</a>, <a href="#HaraguchiShimakawa20">Haraguchi-Shimakawa 20</a> proceeds as follows: <div> <div class="num_prop" id="AdjunctionBetweenTopologicalSpacesAndDiffeologicalSpaces"> <h6 id="proposition">Proposition</h6> (<a class="existingWikiWord" href="/nlab/show/adjunction+between+topological+spaces+and+diffeological+spaces">adjunction between topological spaces and diffeological spaces</a>) There is a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a> <div class="maruku-equation" id="eq:AdjointFunctorsBetweenTopSpAndDifflgSp">(1)<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>TopSp</mi><munderover><mrow><mphantom><mi>AA</mi></mphantom><mo>⊥</mo><mphantom><mi>AA</mi></mphantom></mrow><munder><mo>⟶</mo><mi>Cdfflg</mi></munder><mover><mo>⟵</mo><mi>Dtplg</mi></mover></munderover><mi>DifflgSp</mi></mrow><annotation encoding="application/x-tex"> TopSp \underoverset{ \underset{ Cdfflg }{\longrightarrow} }{ \overset{ Dtplg }{\longleftarrow} }{\phantom{AA}\bot\phantom{AA}} DifflgSp </annotation></semantics></math></div> between the <a class="existingWikiWord" href="/nlab/show/categories">categories</a> of <a class="existingWikiWord" href="/nlab/show/Top">TopologicalSpaces</a> and of <a class="existingWikiWord" href="/nlab/show/DiffeologicalSpaces">DiffeologicalSpaces</a>, where <ul> <li> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cdfflg</mi></mrow><annotation encoding="application/x-tex">Cdfflg</annotation></semantics></math> takes a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the continuous diffeology, namely the diffeological space on the same underlying set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">X_s</annotation></semantics></math> whose plots <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>s</mi></msub><mo>→</mo><msub><mi>X</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">U_s \to X_s</annotation></semantics></math> are the <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> (from the underlying <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> of the <a class="existingWikiWord" href="/nlab/show/domain">domain</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>). </li> <li> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dtplg</mi></mrow><annotation encoding="application/x-tex">Dtplg</annotation></semantics></math> takes a <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a> to the diffeological topology (<a class="existingWikiWord" href="/nlab/show/D-topology">D-topology</a>), namely the <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> with the same underlying set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">X_s</annotation></semantics></math> and with the <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a> that makes all its plots <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>s</mi></msub><mo>→</mo><msub><mi>X</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">U_{s} \to X_{s}</annotation></semantics></math> into <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a>: called the <a class="existingWikiWord" href="/nlab/show/D-topology">D-topology</a>. Hence a <a class="existingWikiWord" href="/nlab/show/subset">subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo>⊂</mo><mo>♭</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">O \subset \flat X</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> in the <a class="existingWikiWord" href="/nlab/show/D-topology">D-topology</a> precisely if for each plot <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>U</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f \colon U \to X</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/preimage">preimage</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">f^{-1}(O) \subset U</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> in the <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>. </li> </ul> Moreover: <ol> <li> the <a class="existingWikiWord" href="/nlab/show/fixed+point+of+an+adjunction">fixed points of this adjunction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">X \in</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Top">TopologicalSpaces</a> (those for which the <a class="existingWikiWord" href="/nlab/show/counit+of+an+adjunction">counit</a> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, hence here: a <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a>) are precisely the <a class="existingWikiWord" href="/nlab/show/Delta-generated+topological+spaces">Delta-generated topological spaces</a> (<a href="Delta-generated+topological+space#AsDTopologicalSpaces">i.e.</a> <a class="existingWikiWord" href="/nlab/show/D-topological+spaces">D-topological spaces</a>): <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mtext>is</mtext><mspace width="thickmathspace"></mspace><mi>Δ</mi><mtext>-generated</mtext><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⇔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>Dtplg</mi><mo stretchy="false">(</mo><mi>Cdfflg</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><munderover><mo>⟶</mo><mo>≃</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>ϵ</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></munderover><mi>X</mi></mrow><annotation encoding="application/x-tex"> X \;\,\text{is}\;\Delta\text{-generated} \;\;\;\;\; \Leftrightarrow \;\;\;\;\; Dtplg(Cdfflg(X)) \underoverset{\simeq}{\;\;\epsilon_X\;\;}{\longrightarrow} X </annotation></semantics></math></div></li> <li> this is an <a class="existingWikiWord" href="/nlab/show/idempotent+adjunction">idempotent adjunction</a>, which exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math>-generated/<a class="existingWikiWord" href="/nlab/show/D-topological+spaces">D-topological spaces</a> as a <a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflective subcategory</a> inside <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a> and a <a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">coreflective subcategory</a> inside all <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>: </li> </ol> <div class="maruku-equation" id="eq:DeltaGeneratedSpacesInIdempotentAdjunction">(2)<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>TopologicalSpaces</mi><munderover><mrow><mphantom><mi>AA</mi></mphantom><mo>⊥</mo><mphantom><mi>AA</mi></mphantom></mrow><munder><mo>⟶</mo><mi>Cdfflg</mi></munder><mover><mo>↩</mo><mrow></mrow></mover></munderover><mi>DTopologicalSpaces</mi><munderover><mrow><mphantom><mi>AA</mi></mphantom><mo>⊥</mo><mphantom><mi>AA</mi></mphantom></mrow><munder><mo>↪</mo><mrow></mrow></munder><mover><mo>⟵</mo><mi>Dtplg</mi></mover></munderover><mi>DiffeologicalSpaces</mi></mrow><annotation encoding="application/x-tex"> TopologicalSpaces \underoverset { \underset{ Cdfflg }{\longrightarrow} } { \overset{ }{\hookleftarrow} } {\phantom{AA}\bot\phantom{AA}} DTopologicalSpaces \underoverset { \underset{ }{\hookrightarrow} } { \overset{ Dtplg }{\longleftarrow} } {\phantom{AA}\bot\phantom{AA}} DiffeologicalSpaces </annotation></semantics></math></div> Finally, these adjunctions are a sequence of <a class="existingWikiWord" href="/nlab/show/Quillen+equivalences">Quillen equivalences</a> with respect to the: <table><thead><tr><th></th><th></th><th></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/model+structure+on+D-topological+spaces">model structure on D-topological spaces</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/model+structure+on+diffeological+spaces">model structure on diffeological spaces</a></td></tr> </tbody></table> <blockquote> Caution: There was a gap in the original proof that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>DTopologicalSpaces</mi><msub><mo>≃</mo> <mi>Quillen</mi></msub><mi>DiffeologicalSpaces</mi></mrow><annotation encoding="application/x-tex">DTopologicalSpaces \simeq_{Quillen} DiffeologicalSpaces</annotation></semantics></math>. The gap is claimed to be filled now, see the commented references <a href="model+structure+on+diffeological+spaces#References">here</a>. </blockquote> </div> Essentially these adjunctions and their properties are observed in <a href="diffeological+space#SYH10">Shimakawa, Yoshida & Haraguchi 2010, Prop. 3.1, Prop. 3.2, Lem. 3.3</a>, see also <a href="D-topology#CSW13">Christensen, Sinnamon & Wu 2014, Sec. 3.2</a>. The model structures and Quillen equivalences are due to <a href="#model+structure+on+Delta-generated+topological+spaces#Haraguchi13">Haraguchi 13, Thm. 3.3</a> (on the left) and <a href="model+structure+on+diffeological+spaces#HaraguchiShimakawa13">Haraguchi-Shimakawa 13, Sec. 7</a> (on the right). <div class="proof"> <h6 id="proof">Proof</h6> We spell out the existence of the <a class="existingWikiWord" href="/nlab/show/idempotent+adjunction">idempotent adjunction</a> <a class="maruku-eqref" href="#eq:DeltaGeneratedSpacesInIdempotentAdjunction">(2)</a>: First, to see we have an <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dtplg</mi><mo>⊣</mo><mi>Cdfflg</mi></mrow><annotation encoding="application/x-tex">Dtplg \dashv Cdfflg</annotation></semantics></math>, we check the hom-isomorphism (<a href="adjoint+functor#eq:HomIsomorphismForAdjointFunctors">here</a>). Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>DiffeologicalSpaces</mi></mrow><annotation encoding="application/x-tex">X \in DiffeologicalSpaces</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>∈</mo><mi>TopologicalSpaces</mi></mrow><annotation encoding="application/x-tex">Y \in TopologicalSpaces</annotation></semantics></math>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">(-)_s</annotation></semantics></math> for the underlying sets. Then a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>, hence a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> of the form <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Dtplg</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> f \;\colon\; Dtplg(X) \longrightarrow Y \,, </annotation></semantics></math></div> is a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>s</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mi>X</mi> <mi>s</mi></msub><mo>→</mo><msub><mi>Y</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">f_s \colon X_s \to Y_s</annotation></semantics></math> of the underlying <a class="existingWikiWord" href="/nlab/show/sets">sets</a> such that for every <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊂</mo><msub><mi>Y</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">A \subset Y_s</annotation></semantics></math> and every <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi \colon \mathbb{R}^n \to X</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/preimage">preimage</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>s</mi></msub><mo>∘</mo><msub><mi>ϕ</mi> <mi>s</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">(f_s \circ \phi_s)^{-1}(A) \subset \mathbb{R}^n</annotation></semantics></math> is open. But this means equivalently that for every such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∘</mo><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">f \circ \phi</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous</a>. This, in turn, means equivalently that the same underlying function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">f_s</annotation></semantics></math> constitutes a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo>˜</mo></mover><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mi>Cdfflg</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\widetilde f \;\colon\; X \longrightarrow Cdfflg(Y)</annotation></semantics></math>. In summary, we thus have a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> of <a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Hom</mi><mo stretchy="false">(</mo><mi>Dtplg</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>Hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Cdfflg</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msub><mi>f</mi> <mi>s</mi></msub></mtd> <mtd><mo>↦</mo></mtd> <mtd><mo stretchy="false">(</mo><mover><mi>f</mi><mo>˜</mo></mover><msub><mo stretchy="false">)</mo> <mi>s</mi></msub><mo>=</mo><msub><mi>f</mi> <mi>s</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Hom( Dtplg(X), Y ) &\simeq& Hom(X, Cdfflg(Y)) \\ f_s &\mapsto& (\widetilde f)_s = f_s } </annotation></semantics></math></div> given simply as the <a class="existingWikiWord" href="/nlab/show/identity+function">identity</a> on the underlying <a class="existingWikiWord" href="/nlab/show/functions">functions</a> of underlying sets. This makes it immediate that this hom-isomorphism is <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> and this establishes the <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a>. Next, to see that the <a class="existingWikiWord" href="/nlab/show/D-topological+spaces">D-topological spaces</a> are the <a class="existingWikiWord" href="/nlab/show/fixed+point+of+an+adjunction">fixed points</a> of this adjunction, we apply the above <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a> on hom-sets to the case <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Hom</mi><mo stretchy="false">(</mo><mi>Dtplg</mi><mo stretchy="false">(</mo><mi>Cdfflg</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>Hom</mi><mo stretchy="false">(</mo><mi>Cdfflg</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Cdfflg</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><msub><mi>ϵ</mi> <mi>Z</mi></msub><msub><mo stretchy="false">)</mo> <mi>s</mi></msub></mtd> <mtd><mo>↦</mo></mtd> <mtd><mo stretchy="false">(</mo><mi mathvariant="normal">id</mi><msub><mo stretchy="false">)</mo> <mi>s</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Hom( Dtplg(Cdfflg(Z)), Y ) &\simeq& Hom(Cdfflg(Z), Cdfflg(Y)) \\ (\epsilon_Z)_s &\mapsto& (\mathrm{id})_s } </annotation></semantics></math></div> to find that the <a class="existingWikiWord" href="/nlab/show/counit+of+the+adjunction">counit of the adjunction</a> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Dtplg</mi><mo stretchy="false">(</mo><mi>Cdfflg</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>ϵ</mi> <mi>X</mi></msub></mrow></mover><mi>X</mi></mrow><annotation encoding="application/x-tex"> Dtplg(Cdfflg(X)) \overset{\epsilon_X}{\longrightarrow} X </annotation></semantics></math></div> is given by the <a class="existingWikiWord" href="/nlab/show/identity+function">identity function</a> on the underlying sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ϵ</mi> <mi>X</mi></msub><msub><mo stretchy="false">)</mo> <mi>s</mi></msub><mo>=</mo><msub><mi>id</mi> <mrow><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>s</mi></msub><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">(\epsilon_X)_s = id_{(X_s)}</annotation></semantics></math>. Therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\eta_X</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, namely a <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a>, precisely if the open subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">X_s</annotation></semantics></math> with respect to the topology on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> are precisely those with respect to the topology on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dtplg</mi><mo stretchy="false">(</mo><mi>Cdfflg</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Dtplg(Cdfflg(X))</annotation></semantics></math>, which means equivalently that the open subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> coincide with those whose pre-images under all continuous functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi \colon \mathbb{R}^n \to X</annotation></semantics></math> are open. This means equivalently that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a D-topological space. Finally, to see that we have an <a class="existingWikiWord" href="/nlab/show/idempotent+adjunction">idempotent adjunction</a>, it is sufficient to check (by <a href="idempotent+adjunction#EquivalentConditionsForIdempotency">this Prop.</a>) that the <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Dtplg</mi><mo>∘</mo><mi>Cdfflg</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>TopologicalSpaces</mi><mo>→</mo><mi>TopologicalSpaces</mi></mrow><annotation encoding="application/x-tex"> Dtplg \circ Cdfflg \;\colon\; TopologicalSpaces \to TopologicalSpaces </annotation></semantics></math></div> is an <a class="existingWikiWord" href="/nlab/show/idempotent+comonad">idempotent comonad</a>, hence that <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Dtplg</mi><mo>∘</mo><mi>Cdfflg</mi><mover><mo>⟶</mo><mrow><mi>Dtplg</mi><mo>⋅</mo><mi>η</mi><mo>⋅</mo><mi>Cdfflg</mi></mrow></mover><mi>Dtplg</mi><mo>∘</mo><mi>Cdfflg</mi><mo>∘</mo><mi>Dtplg</mi><mo>∘</mo><mi>Cdfflg</mi></mrow><annotation encoding="application/x-tex"> Dtplg \circ Cdfflg \overset{ Dtplg \cdot \eta \cdot Cdfflg }{\longrightarrow} Dtplg \circ Cdfflg \circ Dtplg \circ Cdfflg </annotation></semantics></math></div> is a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a>. But, as before for the adjunction counit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math>, we have that also the <a class="existingWikiWord" href="/nlab/show/adjunction+unit">adjunction unit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/identity+function">identity function</a> on the underlying sets. Therefore, this being a natural isomorphism is equivalent to the operation of passing to the D-topological refinement of the topology of a topological space being an idempotent operation, which is clearly the case. </div> </div> <h2 id="References">References</h2> A proof of a model structure on diffeological spaces, with weak equivalences detected on standard smooth homotopy groups, is claimed in <ul> <li id="HaraguchiShimakawa13"><a class="existingWikiWord" href="/nlab/show/Tadayuki+Haraguchi">Tadayuki Haraguchi</a>, <a class="existingWikiWord" href="/nlab/show/Kazuhisa+Shimakawa">Kazuhisa Shimakawa</a>, A model structure on the category of diffeological spaces (<a href="https://arxiv.org/abs/1311.5668">arXiv:1311.5668</a>)</li> </ul> though <a href="#Kihara16">Kihara 16</a> writes (<a href="https://arxiv.org/pdf/1605.06794.pdf#page=2">p. 2</a>) that <blockquote> there exists a gap in the proof of <a href="#HaraguchiShimakawa13">Haraguchi-Shimakawa 13</a>, Theorem 5.6 </blockquote> This gap has been filled in <ul> <li id="Haraguchi18"> <a class="existingWikiWord" href="/nlab/show/Tadayuki+Haraguchi">Tadayuki Haraguchi</a>, Homotopy structures of smooth CW complexes (<a href="https://arxiv.org/abs/1811.06175">arXiv:1811.06175</a>) </li> <li id="HaraguchiShimakawa20"> <a class="existingWikiWord" href="/nlab/show/Tadayuki+Haraguchi">Tadayuki Haraguchi</a>, <a class="existingWikiWord" href="/nlab/show/Kazuhisa+Shimakawa">Kazuhisa Shimakawa</a>, A model structure on the category of diffeological spaces, I, Mathematical Journal of Okayama University (2024) [<a href="https://arxiv.org/abs/2011.12842">arXiv:2011.12842</a>] </li> </ul> A different model structure, however not using the standard smooth homotopy groups(!), is claimed (and published) in <ul> <li id="Kihara16"><a class="existingWikiWord" href="/nlab/show/Hiroshi+Kihara">Hiroshi Kihara</a>, Model category of diffeological spaces, Journal of Homotopy and Related Structures, (2018), 1-40 (<a href="https://arxiv.org/abs/1605.06794">arXiv:1605.06794</a>)</li> </ul> See also <ul> <li id="ChristensenWu14"> <a class="existingWikiWord" href="/nlab/show/J.+Daniel+Christensen">J. Daniel Christensen</a>, <a class="existingWikiWord" href="/nlab/show/Enxin+Wu">Enxin Wu</a>, The homotopy theory of diffeological spaces, I. Fibrant and cofibrant objects, New York J. Math. 20 (2014), 1269-1303 (<a href="http://arxiv.org/abs/1311.6394">arXiv:1311.6394</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Hiroshi+Kihara">Hiroshi Kihara</a>, Quillen equivalences between the model categories of smooth spaces, simplicial sets, and arc-gengerated spaces (<a href="https://arxiv.org/abs/1702.04070">arXiv:1702.04070</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Tadayuki+Haraguchi">Tadayuki Haraguchi</a>, <a class="existingWikiWord" href="/nlab/show/Kazuhisa+Shimakawa">Kazuhisa Shimakawa</a>, On homotopy types of diffeological cell complexes (<a href="https://arxiv.org/abs/1912.05359">arXiv:1912.05359</a>) </li> </ul> </body></html> </div> <div class="revisedby"> Last revised on July 18, 2024 at 08:43:37. 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